On Counting Certain Abelian Varieties Over Finite Fields

Abstract

This paper contains two parts toward studying abelian varieties from the classification point of view. In a series of papers [Doc. Math., 21, 1607–1643 (2016)], [Taiwanese J. Math., 20(4), 723–741 (2016)], etc., the current authors and T. C. Yang obtain explicit formulas for the numbers of superspecial abelian surfaces over finite fields. In this paper, we give an explicit formula for the size of the isogeny class of simple abelian surfaces with real Weil number √q. This establishes a key step that extends our previous explicit calculation of superspecial abelian surfaces to those of supersingular abelian surfaces. The second part is to introduce the notion of genera and idealcomplexes of abelian varieties with additional structures in a general setting. The purpose is to generalize the previous work by the second named author [Forum Math., 22(3), 565–582 (2010)] on abelian varieties with additional structures to similitude classes, which establishes more results on the connection between geometrically defined and arithmetically defined masses for further investigations.